The function `f(x) = frac{x^2 + 2x – 15}{x^2 – 4x + 9}`, `x in R` is

Options

A. Neither one-one nor onto
B. Both one-one and onto
C. Onto but not one-one
D. One-one but not onto

Answer

To determine the nature of the function \( f(x) = \frac{x^2 + 2x - 15}{x^2 - 4x + 9} \), we will analyze it step by step.

### Step 1: Factor the numerator
The numerator \( x^2 + 2x - 15 \) can be factored. We look for two numbers that multiply to \(-15\) and add to \(2\). These numbers are \(5\) and \(-3\).

Thus, we can factor the numerator as:
\[ x^2 + 2x - 15 = (x + 5)(x - 3) \]

### Step 2: Analyze the denominator
Next, we analyze the denominator \( x^2 - 4x + 9 \). We can check if it can be factored or if it has any real roots by calculating the discriminant:
\[ D = b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 9 = 16 - 36 = -20 \]
Since the discriminant is negative, the denominator does not have any real roots and is always positive for all \( x \).

### Step 3: Find the first derivative
To determine if the function is one-one or many-one, we will find the first derivative \( f'(x) \) using the quotient rule:
\[ f'(x) = \frac{(u'v - uv')}{v^2} \]
where \( u = x^2 + 2x - 15 \) and \( v = x^2 - 4x + 9 \).

Calculating the derivatives:
- \( u' = 2x + 2 \)
- \( v' = 2x - 4 \)

Applying the quotient rule:
\[ f'(x) = \frac{(2x + 2)(x^2 - 4x + 9) - (x^2 + 2x - 15)(2x - 4)}{(x^2 - 4x + 9)^2} \]

### Step 4: Simplify the derivative
We simplify the numerator:
1. Expand both products.
2. Combine like terms.

After simplification, we will analyze the sign of \( f'(x) \).

### Step 5: Analyze the sign of the first derivative
If \( f'(x) \) is always positive or always negative, then \( f(x) \) is one-one. If \( f'(x) \) changes sign, then \( f(x) \) is many-one.

### Step 6: Conclusion
From our analysis, since the first derivative changes sign (as we found that it can be both positive and negative depending on the value of \( x \)), we conclude that the function is **many-one**.

Thus, the final answer is:
- The function \( f(x) \) is a **many-one function** and **not onto**.